ThmDex – An index of mathematical definitions, results, and conjectures.
Result R1450 on D1301: Generated subgroup
Explicit algebraic expression for elements of generated subgroup
Formulation 0
Let $G$ be a D22: Group such that
(i) $E \subseteq G$ is a D78: Subset of $G$
(ii) \begin{equation} E \neq \emptyset \end{equation}
(iii) $\langle E \rangle$ is a D1301: Generated subgroup of $G$ with generator $E$
Then \begin{equation} \langle E \rangle = \left\{ g^{n_1}_1 g^{n_2}_2 \cdots g^{n_m}_m \mid m \in 1, 2, 3, \ldots; \, n_1, \ldots, n_m \in \mathbb{Z}; \, g_1, \ldots, g_m \in E \right\} \end{equation}